STAT 280: OPTIMIZATION
SPRING 2017
PROBLEM SET 3
In the following, we write R+ = [0, ∞) and R++ = (0, ∞). So Rn+ = {(x1 , . . . , xn ) ∈ Rn : xi ≥ 0}
and Rn++ = {(x1 , . . . , xn ) ∈ Rn : xi > 0}.
1. (a) Find all stationary points of the cubic polynomial
f (x, y) = x3 + y 3 − 3x − 12y + 20.
Indicate which are the local maximizers and local minimizers.
(b) A rectangular box, open at the top, has a volume of 32 cubic feet. Find the dimensions of
the box so that the total surface area is minimized.
(c) Prove that for any x ≥ 0, y ≥ 0, we always have
x2 + y 2
≤ ex+y−2 .
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(d) Let A ∈ Sn , b ∈ Rn , and c ∈ R. Show that 12 xT Ax + bT x + c has a unique global minimizer
iff A 0. What is it?
(e) Let A ∈ Rm×n and b ∈ Rm . Show that x∗ is a global minimizer of kAx − bk2 iff x∗ is a
solution to AT Ax = AT b.
2. Compute the Hessians of the following functions and decide if they are convex, concave, or
neither on their respective domains.
(a) e, f, g, h : R2++ → R defined by
e(x, y) = xy,
f (x, y) =
1
,
xy
g(x, y) =
x
,
y
h(x, y) = xα y 1−α ,
where α ∈ [0, 1].
(b) ϕ : R × R++ → R defined by ϕ(x, y) = x2 /y. (Hint: Write ∇2 ϕ(x, y) as a rank-1 matrix).
(c) Approximate e, f, g, ϕ by a 3-term Taylor series about the point x = [ 23 ]. Compare the
value of each function at y = [ 2.1
3.2 ] and the value of its corresponding 3-term Taylor series
approximation.
3. (a) Find the Hessian of the function f : Rn → R defined by f (x) = log(ex1 + · · · + exn ). Show
that for any v ∈ Rn ,
X
Xn
X n
2 n
1
xi
2 xi
xi
vT ∇2 f (x)v = x1
e
v
e
−
v
e
.
i
i=1
i=1 i
i=1
(e + · · · + exn )2
Hence or otherwise, deduce that f is a convex function.
(b) Find the Hessian of the function g : Rn++ → R defined by g(x) = (x1 · · · · · xn )1/n . Show
that for any v ∈ Rn ,
"
#
Xn v 2 Xn vi 2
g(x)
i
−
vT ∇2 g(x)v = − 2 n
i=1 x2
i=1 xi
n
i
Hence or otherwise, deduce that g is a concave function.
Date: May 2, 2017 (Version 1.0); due: May 11, 2017.
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STAT 280 ASSIGNMENT 3
(c) Find the Hessian of the function h : Rn++ → R defined by
1
.
h(x) =
1/x1 + · · · + 1/xn
By emulating what we did in the previous two parts or otherwise, decide if h is convex,
concave, or neither on Rn++ .
(d) Find the Hessian of the function ϕ : Rn++ → R defined by ϕ(x) = log h(x). Decide if ϕ is
convex, concave, or neither on Rn++ .